It's unlikely that the parts of the questions that say: "Define as" or "Define to be" are asking you to do anything. I know it sounds like they're telling you to do something, but what those phrases really mean is that they have created a name for something and are telling you what that name means. For example, if I said "Define the function f(x) to equal 1 when x > 0 and 0 when x <=0", then there's nothing for you to do. I'm basically telling you how to interpret expressions such as f(3). When you see f(3), I've just told you that f(x) = 1 when x > 0, therefore f(3) = 1. If I said define a sequence a(n) so that a(0) = 1 and a(n+1)=a(n)+3, I've just told you how to interpret things like a(1), a(2), a(3). So I'm not telling you to actually do anything, I'm just giving you information so that you can understand the next things that I write. We don't really do this in English because we don't really need to be constantly coming up with new names and symbols for things. But if I said to you that from now on when I say the word "ploof", I mean a "fluffy dog", then that would be the same as the mathematic directive: "Define 'ploof' to mean 'fluffy dog'". Note: I'm just guessing here. I could very well be that a question is asking you to come up with your own definitions. I'd have to see the entire question and surrouding context to be sure. More on reddit.com

"Mean" has a precise mathematical definition (and in fact, there are several types) and you're very unlikely to hear it outside those contexts. "Average" is usually a colloquial term that usually means arithmetic mean, but can also mean "typical" or "neither good nor bad." More on reddit.com

Here you are: mean (n.) "that point, place, or state which is halfway between extremes;" c. 1300, originally in music, "a tone intermediate between two other tones," from Old French meien "middle, means, intermediary," noun use of adjective from Late Latin medianus "of or that is in the middle," from Latin medius "in the middle," from PIE root *medhyo- "middle." The modern range of senses in English mostly appeared late 14c.: "course of action; method, way of attaining an end" (as in ways and means;by means of; by all means; by no means); also "the golden mean, moderation;" and "something physically between two extremes." The mathematical sense, "a quantity having a value intermediate between the values of other quantities, the average obtained by adding several quantities together and dividing the sum by their number" is from mid-15c. Some senses reflect confusion with mean (adj.1). The source is the Online Etymological Dictionary , which is a free and semi-scholarly resource for just about all of your etymological needs. More on reddit.com